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Mirrors > Home > ILE Home > Th. List > mulginvinv | GIF version |
Description: The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.) |
Ref | Expression |
---|---|
mulginvcom.b | ⊢ 𝐵 = (Base‘𝐺) |
mulginvcom.t | ⊢ · = (.g‘𝐺) |
mulginvcom.i | ⊢ 𝐼 = (invg‘𝐺) |
Ref | Expression |
---|---|
mulginvinv | ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑁 · (𝐼‘𝑋))) = (𝑁 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulginvcom.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mulginvcom.i | . . . . 5 ⊢ 𝐼 = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 12811 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵) |
4 | 3 | 3adant2 1016 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵) |
5 | mulginvcom.t | . . . 4 ⊢ · = (.g‘𝐺) | |
6 | 1, 5, 2 | mulginvcom 12896 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ (𝐼‘𝑋) ∈ 𝐵) → (𝑁 · (𝐼‘(𝐼‘𝑋))) = (𝐼‘(𝑁 · (𝐼‘𝑋)))) |
7 | 4, 6 | syld3an3 1283 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘(𝐼‘𝑋))) = (𝐼‘(𝑁 · (𝐼‘𝑋)))) |
8 | 1, 2 | grpinvinv 12826 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝐼‘𝑋)) = 𝑋) |
9 | 8 | 3adant2 1016 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝐼‘𝑋)) = 𝑋) |
10 | 9 | oveq2d 5885 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘(𝐼‘𝑋))) = (𝑁 · 𝑋)) |
11 | 7, 10 | eqtr3d 2212 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑁 · (𝐼‘𝑋))) = (𝑁 · 𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5212 (class class class)co 5869 ℤcz 9242 Basecbs 12445 Grpcgrp 12767 invgcminusg 12768 .gcmg 12872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-ltadd 7918 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-2 8967 df-n0 9166 df-z 9243 df-uz 9518 df-seqfrec 10432 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-0g 12655 df-mgm 12667 df-sgrp 12700 df-mnd 12710 df-grp 12770 df-minusg 12771 df-mulg 12873 |
This theorem is referenced by: (None) |
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